Well, I have to admit - there is no proper scientific proof beyond any doubt. Obviously, to prove the statement, one would have to keep two groups of people locked in the lab for years, feed one group with fresh food, and the other group with processed food, and compare the health of both groups at the end.
Then there would be endless arguments: which particular types of food are harmful, and maybe they can be eaten in moderation without any harm at all, and so on, just like the battles that are currently going on around sugar ...
However, there are multiple cases where the introduction of "Western foods" to people who were eating traditional diets caused significant increase in "diseases of civilisation". I personally started from reading "Nutrition and Physical Degeneration" by Weston A. Price - it is available for free online.
To generalise more, unprocessed foods have more fibre. Increased fibre is proven to reduce cancer incidence; eg[1].
(Of course, one can eat processed food and add fibre from another source. Nevertheless, the advice to eat unprocessed food is reasonable as a generalisation).
This is a sweeping generalisation that I'd like to see evidence for.